First Variations of the Best Sobolev Trace Constant with Respect to the Domain

Abstract

AbstractIn this paper we study the best constant of the Sobolev trace embedding H1(Ω) → L2(∂Ω), where Ω is a bounded smooth domain in ℝN. We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.